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3. Let V be a normed linear space and W a subspace of V . Let f V . Prove that the set of best
3. Let V be a normed linear space and W a subspace of V . Let f V . Prove that the set of best approximations to f by elements in W is a convex set.
4. Let f and g in C[0,1], and constants, and denote by Bnf the Bernstein polynomial of f of degree n. Prove that
(a) Bn(f + g) = Bnf + Bng, i.e. Bn is a linear operator in C[0,1]. (b) If f(x) g(x) for all x [0,1] then Bnf(x) Bng(x) for all x [0,1], i.e. Bn is a monotone operator.
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